Astronomical Observations on the Planets Venus and Mars, Made with a View to Determine the Heliocentric Longitude of Their Nodes, the Annual Motion of the Nodes, and the Greatest Inclination of Their Orbits. By Thomas Bugge, F. R. S. Regius Professor of Astronomy at Copenhagen, Member of the Academies of Stockholm, Copenhagen, Manheim, and Drontheim, and Correspondent of the Academy of Sciences at Paris

II. Astronomical Observations on the Planets Venus and Mars, made with a View to determine the heliocentric Longitude of their Nodes, the annual Motion of the Nodes, and the greatest Inclination of their Orbits. By Thomas Bugge, F. R. S. Regius Professor of Astronomy at Copenhagen, Member of the Academies of Stockholm, Copenhagen, Manheim, and Drontheim, and Correspondent of the Academy of Sciences at Paris. Read November 26, 1789. I. The heliocentric longitude and annual motion of Venus's nodes. The following astronomical observations were made at the Royal Observatory at Copenhagen with a six-feet transit instrument, and with a mural quadrant of six-feet radius. It would be too tedious to enumerate all the original observations, which either are already printed in the first volume of my Astronomical Observations, or will very soon be published in the second volume. I only shall set down the observed geocentric longitudes and latitudes, corrected for aberration and nutation, and compared with the tables of Dr. Halley and of M. de la Lande. | Mean time at Copenhagen | Observed geocentric longitude of ♈ | Observed geocentric latitude of ♈ | Halley's error, in long. | Halley's error, in lat. | De la Lande's error, in long. | De la Lande's error, in lat. | |------------------------|-----------------------------------|----------------------------------|-------------------------|-------------------------|-----------------------------|-----------------------------| | 1781 | | | | | | | | Sept. 13 | 1° 38' 6" 6° 18' 39" 15" | 3° 28' 12" N | +11" | +3" | | | | 22 | 1° 43' 24" 6° 29' 38" 48" | 3° 3' 9" | +17" | +1" | +39" | +3" | | Oct. | 1° 49' 46" 7° 10' 36" 42" | 0° 23' 48" 8" | +16" | -7" | +44" | -2" | | | 1° 52' 11" 7° 14' 15" 17" | 0° 33' 3" | -12" | -4" | | | | 1784 | | | | | | | | Sept. 20 | 0° 37' 17" 6° 9' 40" 27" | 1° 6' 15" N | +41" | +15" | +17" | +12" | | 25 | 0° 39' 57" 6° 15' 52" 59" | 0° 57' 50" | +15" | +3" | | | | Oct. | 2° 44' 49" 6° 24' 35" 34" | 0° 44' 21" | +36" | +6" | | | | | 1° 54' 7" 7° 9' 30" 40" | 0° 16' 41" | +10" | +1" | | | | | 1° 0' 42" 7° 18' 13" 11" | 0° 1' 3" 8" | +17" | +1" | +18" | +3" | | 1786 | | | | | | | | Aug. 19 | 2° 25' 57" 6° 4' 45" 54" | 0° 23' 7" N | -5" | -3" | | | | | 2° 20' 14" 6° 5' 56" 11" | 0° 19' 40" | -17" | -9" | | | | | 2° 26' 32" 6° 7' 6" 42" | 0° 16' 20" | -13" | -5" | +17" | -12" | | | 2° 28' 30" 6° 15' 7" 5" | 0° 8' 46" S | -0" | -2" | +23" | +2" | | | 2° 28' 47" 6° 16' 27" 39" | 0° 12' 32" | -22" | -4" | | | The angle at the planet or the commutation \(= P\) is not directly to be taken out of the table. The difference between the observed geocentric longitude of the planet and the geocentric longitude of the sun, calculated from M. Mayer's solar tables, is the angle at the earth, or the elongation \(= T\). From this elongation, which is to be depended upon to very few seconds, and from the planet's and the earth's distances from the sun, according to the tables, the commutation is calculated, and the geocentric longitude is reduced to the heliocentric longitude. The angles \(P\), \(T\), and \(S\), at the sun, thus found, are likewise used to calculate the heliocentric latitude. According According to the different dimensions, given to the orbit of the planet in the different tables, the radius vector at a given time will also be somewhat different. These differences in the tables of Dr. Halley and M. de la Lande are but small: thus, 1784, September 20, at 0 h. 37' 17'' mean time, the observed and apparent geocentric longitude of Venus 6 s. 9° 39' 54'', the aberration and nutation +33'', the corrected and true geocentric longitude 6 s. 9° 40' 27'', the sun's geocentric longitude 5 s. 28° 5' 51''; hence the elongation $T = 0 s. 11° 34' 36''$. If now the logarithm of the radius vector SP is taken out of Dr. Halley's tables = 4.858251, then $\sin P = \frac{ST \times \sin T}{SP}$ and $P = 16° 11' 52''$; but if the logarithm SP is taken out of the tables of M. de la Lande = 4.858168, the angle P will be found = 16° 12' 4'', the difference is 12''. This uncertainty in the commutation, and consequently in the heliocentric longitude, would have been still greater if the calculations had been made only from the tables, or from the planet's geocentric longitude by the tables; thus this angle P is, according to Dr. Halley, = 16° 10' 52''; and from the tables of M. de la Lande = 16° 10' 13''. Let the difference between two heliocentric longitudes, one before and the other after the passage through the node, be \(a\), the northern heliocentric latitude \(=b\), and the southern \(=\beta\); let the arc of the ecliptic from the node to the longitude, answering to the southern latitude, be \(x\); then \[ \tan x = \frac{\sin a \cdot \tan \beta}{\tan b + \cot a \tan \beta}. \] By this formula the distance from every longitude with a southern latitude to the node may be found; and hence the heliocentric longitude of the node. --- ### Observations | Mean time at Copenhagen | Heliocentric longitude of \(q\) in the ecliptic | Heliocentric latitude of \(q\) | Halley's error, in long. | Halley's error, in lat. | De la Lande's error, in long. | De la Lande's error, in lat. | |------------------------|-----------------------------------------------|-------------------------------|--------------------------|-------------------------|-------------------------------|-----------------------------| | 1781 | | | | | | | | Sept. 13 | 1 h 38 m 6 s 28 d 40 m 26 s | 0° 56' 13" N | +2" | +5" | | | | | 22° 43' 24" | 0° 6' 4" | +70" | +2" | +63" | -6" | | Oct. | 1 h 49 m 46 s 27 d 15 m 26 s | 0° 44' 6" S | +41" | -13" | +13" | -6" | | | 4° 52' 11" | 1° 0' 28" | -20" | -8" | | | | 1784 | | | | | | | | Sept. 20 | 0 h 37 m 17 s 25 d 51 m 19 s | 2° 23' 46" N | +28" | +34" | +106" | +27" | | | 25° 39' 57" | 2° 13' 7" | +1" | +7" | | | | Oct. | 2 h 44 m 49 s 15 d 6 m 32" | 1° 40' 46" | +74" | +13" | | | | | 14° 54' 7" | 0° 36' 58" | +33" | +1" | | | | | 21° 0' 42" | 0° 2' 17" S | +55" | -2" | +43" | +7" | | 1786 | | | | | | | | Aug. 19 | 2 h 25 m 57 s 4 d 12 m 41 s | 0° 37' 6" N | +25" | -4" | | | | | 20° 26' 14" | 0° 31' 22" | +52" | -15" | | | | | 21° 26' 32" | 0° 25' 55" | +68" | -6" | +69" | -16" | | | 28° 28' 30" | 0° 13' 17" S | +108" | -3" | +89" | +3" | | | 29° 28' 47" | 0° 18' 52" | +35" | -6" | | | | Observations compared | Heliocentric longitude of SS S. | Reduced to 1786 | |-----------------------|---------------------------------|----------------| | **1781** | | | | September 13 — October 1 | 8 14 42 42 | | | September 13 — October 4 | 8 14 42 23 | | | September 22 — October 1 | 8 14 42 29 | | | September 22 — October 4 | 8 14 42 12 | | | Mean | 8 14 42 24 | 8 14 44 53 | | **1784** | | | | October 21 — September 20 | 8 14 43 38 | | | October 21 — September 25 | 8 14 43 40 | | | October 21 — October 2 | 8 14 43 42 | | | October 21 — October 4 | 8 14 43 30 | | | Mean | 8 14 43 38 | 8 14 44 34 | | **1786** | | | | August 19 — August 29 | 8 14 45 3 | | | August 19 — August 28 | 8 14 44 28 | | | August 20 — August 29 | 8 14 44 16 | | | August 20 — August 28 | 8 14 44 0 | | | August 21 — August 29 | 8 14 44 27 | | | August 21 — August 28 | 8 14 44 36 | | | Mean | 8 14 44 28 | 8 14 44 28 | Hence the heliocentric longitude of the descending node of the planet Venus was, 1786, August 25, at 8 h. 39′ = 8 s. 14° 44′ 38″. I think that this place of the node is to be depended upon to 10 or 15 seconds. According to the tables of M. Cassini, the longitude of the node 8 s. 14° 48′ 31″, the difference, or the error, −3′ 53″. According to the tables of Dr. Halley 8 s. 14° 42′ 39″, the difference +1′ 59″. From the tables of M. de la Lande 8 s. 14° 45′ 15″, the difference only −37″. Vol. LXXX. E In In order to ascertain the annual motion of the node, this observation is to be compared with the observations of other astronomers. The numbers in the column A are found by setting out from my own observations 1786. In the column B, the observation of M. de la Lande 1769 is taken as the first; the series C is begun with the determination of M. Horrox 1639; and in the series D, M. Cassini's observation 1698 is taken as the basis. | The Astronomer | The time of observation | Heliocentric longitude of Θ Φ | Annual motion of Θ Φ | |----------------|-------------------------|-------------------------------|----------------------| | Horrox | 1639, Dec. | s. o 13 27 50 | 31,3 31,6 - - | | Cassini | 1698, Sept. | 42 14 1 45 | 29,2 29,2 36,5 | | Cassini | 1705, June | 12 14 2 52 | 30,9 31,3 31,9 | | Cassini | 1731, April | 7 14 17 2 | 30,0 30,2 32,7 24,9 | | de la Caille | 1746, Dec. | 21 14 23 10 | 32,1 34,3 31,0 26,8 | | de la Caille | 1761, June | 5 14 31 30 | 31,2 - - 31,3 27,3 | | de la Lande | 1769, June | 3 14 36 20 | 28,1 - - 31,6 29,2 | | Bugge | 1786, Aug. | 2 14 44 38 | - - - - 31,3 29,2 | Mean 30,4 31,3 32,3 27,5 If the mean be taken of those four means, the annual motion of Venus's node will be 30,37'', or very near 31'', adopted in the tables of Dr. Halley and M. de la Lande. II. The greatest inclination of the orbit of Venus to the ecliptic. In the first place, I shall put down the observed geocentric longitudes and latitudes, corrected for aberration and nutation. | Mean time at Copenhagen | Geocentric longitude of ψ | Geocentric latitude of ψ | Halley's error, in long., in lat. | De la Lande's error, in long., in lat. | |------------------------|--------------------------|--------------------------|----------------------------------|-------------------------------------| | | h. | s. | | | | 1781, July 20 | 0 1 40 | 11 10 49 | + 3° | + 5 | | | 24 5 45 | 16 5 52 | + 21 | + 5 | | | 30 1 11 18 | 23 28 41 | + 29 | + 9 | | Aug. | 31 1 12 9 | 24 42 20 | + 32 | + 12 | | | 4 1 12 58 | 25 55 54 | + 18 | + 3 | | | 15 21 4 | 29 37 13 | + 20 | + 7 | | 1782, July 13 | 21 9 19 | 10 54 17 | + 53 | + 2 | | Nov. | 5 22 50 | 43 6 29 46 | + 40 | + 3 | | | | 25 1 21 11 | | + 65 | | | | | | + 4 | | 1783, Sept. 19 | 2 4 37 | 3 52 29 | - 115 | + 39 | | | 20 2 2 7 | 4 15 51 | - 122 | + 49 | | | 26 1 43 58 | 5 55 9 | - 190 | + 55 | | Oct. | 2 1 22 55 | 6 20 38 | - 130 | + 53 | | | | | | - 24 | | | | | | + 44 | | 1784, May 18 | 22 29 58 | 1 7 1 56 | + 57 | + 6 | | Sept. | 8 0 30 11 | 24 45 11 | + 27 | + 3 | | | 20 0 37 17 | 9 40 27 | + 41 | + 15 | | | 25 0 39 57 | 15 52 59 | + 15 | + 3 | | | | | | + 77 | | | | | | + 12 | | 1785, July 29 | 20 53 21 | 2 22 5 27 | - 40 | + 13 | | Nov. | 27 21 29 57 | 9 30 40 | - 5 | + 4 | | | | | | + 26 | | | | | | - 2 | | 1786, June 19 | 1 43 7 | 21 39 32 | + 12 | + 12 | | | 24 1 49 20 | 27 44 41 | + 53 | + 19 | | | 29 1 55 7 | 3 47 42 | + 3 | + 5 | | July | 1 1 57 17 | 6 13 9 | + 9 | + 7 | | | 14 2 9 7 | 21 54 24 | - 6 | + 10 | | | | | | + 43 | | | | | | + 8 | | 1788, May 6 | 3 1 24 | 0 23 44 | + 20 | + 22 | | | 7 3 2 17 | 1 28 48 | + 27 | + 17 | | | 9 3 4 43 | 3 38 1 | + 39 | + 14 | The angle at the planet is found in the manner before mentioned. The following table contains the heliocentric longitudes and latitudes to the moments of mean time in the foregoing table. The heliocentric place of the node is ascertained with a tolerable degree of accuracy; hence the arc of the ecliptic from the node to the circle of latitude, passing through the planet, is given \(=d\); the inclination of the orbit to the ecliptic \(=y\) is to be calculated by this formula, \(\cot. y = \frac{\text{fin. } d}{\text{fin. lat. hel.}}\). | Heliocentric longitude of \(q\) in the ecliptic. | Heliocentric latitude of \(q\). | Halley's error, in long. | in lat. | De la Lande's error, in long. | in lat. | Inclination of \(q\)’s orbit. | |---|---|---|---|---|---|---| | **1781, July 20** | | | | | | | | s. o o 53°3'16"55N | +18°+12" | | | | | 3°23'32" | | 24 | 5°6'30"59°3'21"27" | +7°+7" | | | | 3°23'31" | | 30 | 5°16'15"59°3'23"36" | +12°+21" | +115°+21" | | | 3°23'41" | | 31 | 5°17'53"28°3'23"30" | +16°+20" | | | | 3°23'49" | | Aug. 1 | 5°19'30"31°3'22"43" | +3°+7" | | | | 3°23'27" | | 4 | 5°24'22"35°3'20"41" | +4°+15" | | | | 3°23'35" | | **1782, July 13** | | | | | | | | Nov. 5 | 0°4'2"37°3'11"57"S | +116°+4" | | | | 3°23'26" | | 6°9'46"43°4'18"N | +40°+5" | | | | | 3°23'27" | | **1783, Sept. 19** | | | | | | | | 20 | 11°6'41"21°3'21"45"S | -97°+23" | -183°+23" | | | 3°23'44" | | 26 | 11°17'45"5°3'23"30" | -105°+27" | | | | 3°23'48" | | Oct. 2 | 11°27'16"44°3'18"56" | -117°+31" | -195°+30" | | | 3°23'47" | | **1784, May 18** | | | | | | | | Sept. 8 | 0°5'34"53°3'10"14"S | +71°+13" | -1°+11" | | | 3°23'34" | | 20 | 6°6'30"11°3'8"56"N | +18°+7" | | | | 3°23'27" | | 25 | 7°3'52"40°2'13"7" | +1°+7" | | | | 3°23'58" | | **1785, July 29** | | | | | | | | Nov. 27 | 11°15'44"9°3'23"30"S | -68°+12" | -154°+12" | | | 3°23'32" | | 6°0'50"45°3'15"22"N | +1°+2" | +99°-2" | | | | 3°23'22" | | **1786, June 19** | | | | | | | | 24 | 4°25'53"12°3'12"43"N | +1°+26" | | | | 3°23'41" | | 29 | 5°4'2"2°3'20"31" | +44°+43" | | | | 3°24'0" | | July 1 | 5°12'8"55°3'23"18" | -5°+11" | | | | 3°23'30" | | 14 | 6°6'28"2°3'9"10" | +1°+14" | +109°+14" | | | 3°23'35" | | **1788, May 6** | | | | | | | | 7 | 5°16'58"34°3'23"37"N | +20°+20" | | | | 3°23'46" | | 9 | 5°18'36"3°3'23"12" | +27°+21" | | | | 3°23'28" | | | 5°21'50"56°3'22"4" | +39°+19" | | | | 3°23'38" | Mean 3°23'37.7" The heliocentric latitudes observed 1781 July 30, 1783 September 26, 1785 July 29, 1786 July 1, 1788 May 6, are very near the greatest latitude; the mean of the inclinations found on these days is $3^\circ 23' 40''$, and very near the mean of all the observations $3^\circ 23' 37''$. The inclination, or the greatest heliocentric latitude, may also be found by interpolation of the maximum amongst the observed heliocentric latitudes. This maximum is found 1781 = $3^\circ 23' 39''$, 1783 = $3^\circ 23' 41''$, 1786 = $3^\circ 23' 36''$; the mean of those three maximums $3^\circ 23' 38''$, which inclination may be depended upon to 1 or 2 seconds. The inclination of the orbit of Venus has been supposed in the tables of M. Cassini, Dr. Halley, and M. de la Lande, = $3^\circ 23' 20''$, and the error of the tables + $18''$, 6. III. The heliocentric longitude and motion of the nodes of Mars. In a Paper printed in the Memoirs of the Royal Academy of Sciences at Stockholm, I have determined the heliocentric longitude of Mars's ascending node = 1 s. $17^\circ 54' 24''$, in the year 1783, December 7, 20 h. $23' 39''$, mean time at Copenhagen: the error of M. Cassini's tables - $10' 35''$, of Dr. Halley's tables - $23' 27''$, of M. de la Lande's tables - $4' 37''$. I refer the reader to that Paper (Kongliga Svenska Vetenskaps Academiens nya Handlingar, Tom. VI. for the year 1785, p. 285—290). The annual motion of Mars's node may be found by comparing the following observations of the longitude of the node. In the column A the numbers are going upwards from the observation 1783; in the column B the numbers are going downwards from the observation 1595. | The name of the astronomer | Time of observation | Heliocentric longitude of $g$ & $\delta$ | Annual motion | |---------------------------|--------------------|----------------------------------------|---------------| | Tycho Brahe | 1595, Oct. 28 | $s.$ $16^{\circ}24'33''$ | $28''7$ | | Cassini | 1700, May 6 | $17^{\circ}13'43''$ | $28''9$ | | Cassini | 1721, Nov. 13 | $17^{\circ}29'49''$ | $23''8$ | | de la Caille | 1747, May 14 | $17^{\circ}37'11''$ | $27''5$ | | de la Caille | 1753, Nov. 4 | $17^{\circ}42'5''$ | $24''6$ | | Maskelyne | 1778, April 17 | $17^{\circ}51'40''$ | $28''5$ | | Bugge | 1783, Dec. 7 | $17^{\circ}54'24''$ | $28''7$ | Mean $27''0$ $29''5$ The mean of the two series A and B will give the most probable annual motion of Mars's node $28''$, in the tables of M. Cassini the annual motion is $34''$, in the tables of Dr. Halley $38''$, and in the tables of M. de la Lande $40''$. IV. The inclination of the orbit of the planet Mars. | Mean time at Copenhagen | Geocentric longitude of $\delta$ | Geocentric latitude of $\delta$ | Error of the tables of M. de la Lande, in long. | in lat. | |-------------------------|---------------------------------|---------------------------------|-----------------------------------------------|--------| | 1788, Jan. | | | | | | h. | s. | | | | | 9 11 57 7 | $3^{\circ}16'27''7$ | $4^{\circ}5'25''N$ | $+17$ | $+1$ | | 10 11 51 28 | $3^{\circ}16'35''4$ | $4^{\circ}6'10''$ | $+19$ | $+7$ | | 11 13 45 50 | $3^{\circ}15'40''15$ | $4^{\circ}6'44''$ | $+23$ | $+10$ | | 26 10 24 50 | $3^{\circ}10'43''39$ | $4^{\circ}3'59''$ | $+45$ | $+19$ | | Feb. 14 | $3^{\circ}8'13''29$ | $3^{\circ}39'35''$ | $-8$ | $+8$ | | Mar. 9 | $3^{\circ}11'35''3$ | $3^{\circ}1'13''$ | $-7$ | $-4$ | | 12 7 29 7 | $3^{\circ}11'59''16$ | $2^{\circ}56'52''$ | $+2$ | $+7$ | | 13 7 26 19 | $3^{\circ}12'15''21$ | $2^{\circ}55'24''$ | $+2$ | $+9$ | | 14 7 23 35 | $3^{\circ}12'31''44$ | $2^{\circ}53'56''$ | $+3$ | $+8$ | | 16 7 18 10 | $3^{\circ}13'5'38$ | $2^{\circ}50'56''$ | $-18$ | $+7$ | | April 6 | $3^{\circ}20'29''42$ | $2^{\circ}22'46''7$ | $-17$ | $+8$ | The geocentric longitudes of Mars are corrected for aberration and nutation, and compared with M. de la Lande's newest newest tables, which after the last improvements commonly give the true place of Mars within the fourth part of a minute. The error in longitude $+17''$ signifies that the longitude in the tables is $17''$ too small; and that those $17''$ are to be added to the calculated longitude, in order to make it agree with the observed longitude. | Mean time at Copenhagen | Heliocentric longitude of $\delta$ | Heliocentric latitude of $\delta$ | Error of the tables of M. de la Lande, in long. | Inclination of the orbit of $\delta$. | |------------------------|-----------------------------------|----------------------------------|-----------------------------------------------|--------------------------------------| | 1788 | | | | | | Mar. 9 37 35 | 4 15 12 | 28 1 50 49,1 N | $-7$ | $-3$ | | 12 7 29 | 7 4 16 | 31 42 1 50 53,5 | $+2$ | $-4,5$ | | 13 7 26 | 9 4 16 | 58 8 1 50 55,0 | $+2$ | $-3$ | | 14 7 23 | 35 4 17 | 24 28 1 50 57,7 | $+3$ | $-1,3$ | | 16 7 18 | 19 4 18 | 16 50 56,7 | $-4$ | $-2,3$ | The inclination of Mars is taken in the tables of M. Cassini $1^\circ 50' 54''$, and in the tables of M. de la Lande and Dr. Halley $1^\circ 51' 0''$. I shall conclude this Paper with the opposition of Mars according to the foregoing observations. *The opposition of Mars to the sun happened 1788, January 7, at 8 h. 19' 32'' true time; the apparent geocentric longitude of Mars at that moment = 3 s. 17° 17' 8'', and the geocentric latitude = 4° 4' 3' N.* Saturn was in opposition to the sun, August 29, 20 h. 51' 11'' true time; the apparent longitude $\beta = 11$ s. 7° 31' 34'', and latitude 1° 59' 33'' S. The new planet was in opposition to the sun January 18, o h. 28' 33'' true time, the longitude = 3 s. 28° 10' 7'', and latitude 0° 34' 35'' N.